Question

If `tan x + tan(π/3 - x) tan ((2π)/3 + x)` = 3, then ______.

Options

• tan x = 1

• tan 2x = 1

• tan 3x = 1

• None of these

Answer 1

If `tan x + tan(π/3 - x) tan ((2π)/3 + x)` = 3, then tan 3x = 1.

Explanation:

`tan x + tan(π/3 + x) + tan ((2π)/3 + x)`

= `tan x + (tan x + sqrt(3))/(1 - sqrt(3) tan x) + (tan x - sqrt(3))/(1 + sqrt(3) tan x)`

= `tan x + (8 tan x)/(1 - 3 tan^2 x)`

= `(3(3 tan x - tan^3 x))/(1 - 3 tan^2 x)`

= 3 tan 3x

Therefore, the given equation is 3 tan 3x = 3

`\implies` tan 3x = 1